3.168 \(\int \frac{e^{\frac{1}{x}} (1+x)}{x^4} \, dx\)

Optimal. Leaf size=27 \[ -\frac{e^{\frac{1}{x}}}{x^2}-e^{\frac{1}{x}}+\frac{e^{\frac{1}{x}}}{x} \]

[Out]

-E^x^(-1) - E^x^(-1)/x^2 + E^x^(-1)/x

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Rubi [A]  time = 0.157809, antiderivative size = 27, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 3, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25 \[ -\frac{e^{\frac{1}{x}}}{x^2}-e^{\frac{1}{x}}+\frac{e^{\frac{1}{x}}}{x} \]

Antiderivative was successfully verified.

[In]  Int[(E^x^(-1)*(1 + x))/x^4,x]

[Out]

-E^x^(-1) - E^x^(-1)/x^2 + E^x^(-1)/x

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (x + 1\right ) e^{\frac{1}{x}}}{x^{4}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(exp(1/x)*(1+x)/x**4,x)

[Out]

Integral((x + 1)*exp(1/x)/x**4, x)

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Mathematica [A]  time = 0.00798998, size = 16, normalized size = 0.59 \[ e^{\frac{1}{x}} \left (-\frac{1}{x^2}+\frac{1}{x}-1\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[(E^x^(-1)*(1 + x))/x^4,x]

[Out]

E^x^(-1)*(-1 - x^(-2) + x^(-1))

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Maple [A]  time = 0.001, size = 18, normalized size = 0.7 \[ -{\frac{ \left ({x}^{2}-x+1 \right ){{\rm e}^{{x}^{-1}}}}{{x}^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(exp(1/x)*(1+x)/x^4,x)

[Out]

-(x^2-x+1)*exp(1/x)/x^2

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Maxima [A]  time = 1.42257, size = 23, normalized size = 0.85 \[ -\Gamma \left (3, -\frac{1}{x}\right ) + \Gamma \left (2, -\frac{1}{x}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x + 1)*e^(1/x)/x^4,x, algorithm="maxima")

[Out]

-gamma(3, -1/x) + gamma(2, -1/x)

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Fricas [A]  time = 0.20705, size = 23, normalized size = 0.85 \[ -\frac{{\left (x^{2} - x + 1\right )} e^{\frac{1}{x}}}{x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x + 1)*e^(1/x)/x^4,x, algorithm="fricas")

[Out]

-(x^2 - x + 1)*e^(1/x)/x^2

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Sympy [A]  time = 0.080872, size = 14, normalized size = 0.52 \[ \frac{\left (- x^{2} + x - 1\right ) e^{\frac{1}{x}}}{x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(exp(1/x)*(1+x)/x**4,x)

[Out]

(-x**2 + x - 1)*exp(1/x)/x**2

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \mathit{undef} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x + 1)*e^(1/x)/x^4,x, algorithm="giac")

[Out]

undef